scholarly journals A Relation between Laplace and Hankel Transforms

1962 ◽  
Vol 5 (3) ◽  
pp. 114-115 ◽  
Author(s):  
B. R. Bhonsle
Keyword(s):  
Laplace Transform ◽  
Hankel Transform ◽  
Laplace And Hankel Transforms ◽  
Hankel Transforms ◽  
The Laplace Transform ◽  
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The Laplace transform of a function f(t) ∈ L(0, ∞) is defined by the equationand its Hankel transform of order v is defined by the equationThe object of this note is to obtain a relation between the Laplace transform of tμf(t) and the Hankel transform of f(t), when ℛ(μ) > − 1. The result is stated in the form of a theorem which is then illustrated by an example.

On a generalized L–H transform

1968 ◽  
Vol 64 (2) ◽  
pp. 399-406 ◽  
Author(s):  
V. K. Kapoor ◽  
S. Masood
Keyword(s):  
Laplace Transform ◽  
Characteristic Property ◽  
Hankel Transform ◽  
Inversion Theorem ◽  
The Laplace Transform ◽  
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AbstractThe authors, while attempting to give a new generalization of the Laplace transform, have come across a particular function in the form of Meijer's G-functionwhich, when taken as a nucleus of the transformation K(x) indefines a new transform. This transform besides serving as a generalization of the Laplace transform and most of its generalizations existing in the literature bears the characteristic property of generalizing the Hankel transform (and hence some of its generalizations) as well. In this paper the authors after having defined the transform have given an inversion theorem which is supported by means of two examples.

On Some Properties of Functions Analytic in a Half-Plane

1959 ◽  
Vol 11 ◽  
pp. 432-439 ◽  
Author(s):  
P. G. Rooney
Keyword(s):  
Laplace Transform ◽  
Half Plane ◽  
The Laplace Transform ◽  
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The spaces , w real, 1 ≤ p < ∞, consist of those functions f(s), analytic for Re s > w, and such that μp(f;x) is bounded for x > w, where1.1Doetsch (1) has shown that if e-wtϕ(t) ∈ Lp (0, ∞), 1 < p ≤ 2, and f is the Laplace transform of ϕ, that is,then f ∈ , where1.2and that conversely if f ∈ , 1 < p ≤ 2, then there is a function ϕ, with e-wtϕ(t) ∈ Lq (0, ∞), such that f is the Laplace transform of ϕ.

The Inverse Laplace Transform of the Product of Two Whittaker Functions

1962 ◽  
Vol 58 (4) ◽  
pp. 580-582 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Positive Integer ◽  
Laplace Transform ◽  
Whittaker Function ◽  
Inverse Laplace Transform ◽  
Whittaker Functions ◽  
Original Function ◽  
The Laplace Transform ◽  
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The object of this paper is to obtain the original function of which the Laplace transform (l) is the productwhere, as usual, p is complex, n is any positive integer, and Wk, m(z) is the Whittaker function defined by the equationIn § 2 it will be shown that this original function iswhere the symbol Δ(n; α) represents the set of parameters

A proof of characterisation of oscillation for higher-order neutral differential equations of mixed type by the Laplace transform

1994 ◽  
Vol 124 (5) ◽  
pp. 909-916 ◽  
Author(s):  
O. Arino ◽  
M. A. El Attar
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
General Expression ◽  
Exponential Growth ◽  
Mixed Type ◽  
Deviating Arguments ◽  
Neutral Differential Equations ◽  
Equations Of Mixed Type ◽  
The Laplace Transform ◽  
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Consider the general expression of such equations in the formwhere Ai, Bj, ∊ ℝ, δo = 0 dn/ 0, dn are n-derivatives, n ≧ l, the σj'S and δj,'s respectively, are ordered as an increasing family with possibly positive and negative terms. These are the deviating arguments. In this paper, we provide a proof of this result based on the use of the Laplace transform. Our method involves new results regarding the exponential growth of positive solutions for such equations.

scholarly journals A generalisation of Dirichlet's multiple integral

1956 ◽  
Vol 40 ◽  
pp. 16-18
Author(s):  
S. K. Lakshmana Rao
Keyword(s):  
Laplace Transform ◽  
Mellin Transform ◽  
Multiple Integral ◽  
Real Numbers ◽  
General Integral ◽  
The Laplace Transform ◽  
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The well-known multiple integralwhere Rn is the region defined by x1 ≥ 0, x2 ≥ 0, …., xn ≥ 0, x1 + x2 + …. + xn ≤ 1, and where a0, a1, …, an are positive constants, can be evaluated either in the classical way using the Dirichlet transformation or by the use of the Laplace transform. I. J. Good has considered a more general integral and has proved the following result by induction:—If f1(t), f2(t), …, fn(t) are Lebesgue measurable for 0 ≤ t ≤ 1, m1, m2, …., mn, mn+1 (= 0) are real numbers, Mr = m1 + m2 + … + mr, x1, x2, …, xn are non-negative variables and Xr = x1 + x2 + … + xr, thenIt does not seem to be possible to establish this relation by employing the Laplace transform, but we show below that it can be obtained using the Mellin transform.

scholarly journals The ruin probability in a special case

1971 ◽  
Vol 6 (1) ◽  
pp. 66-68 ◽  
Author(s):  
H. Bohman
Keyword(s):  
Numerical Analysis ◽  
Laplace Transform ◽  
Ruin Probability ◽  
Claim Amount ◽  
The Laplace Transform ◽  
Desk Calculator ◽  
Numerical Tools ◽  
The Mean ◽  
Image Position ◽  
Special Case

It is fantastic how the computer has changed our attitude to numerical problems. In the old days when our numerical tools were paper, pencil, desk calculator and logarithm tables we had to stay away from formulas and methods which led to too lengthy calculations. A consequence is that we have a tendency to think of numerical analysis in terms of the classical tools. If we go back to the results of earlier writers it seems, however, very likely that many results and formulas developed by them which had earlier a theoretical interest only could nowadays be applied successfully in numerical analysis.As an example I take the ruin probability ψ(x). The Laplace transform of ψ(x) is given by the following expressionwhere c > 1. In fact (c — 1) is equal to the “security loading”. The function p(y) is equal to the Laplace transform of the claim distribution. We assume that the mean claim amount is equal to one, i.e. p′(0) = — 1.In his book from 1955 [1] Cramer points out that this formula will be more easy to handle if the claim distribution is an exponential polynomial. In this case we havewhereCramér's results are given on pages 81-83 in his book. We reproduce them here with a slight change of notations only.

scholarly journals The inversion of a transform related to the Laplace transform and to heat conduction

1964 ◽  
Vol 4 (1) ◽  
pp. 1-14 ◽  
Author(s):  
David V. Widder
Keyword(s):  
Heat Conduction ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Laplace Transform ◽  
Physical Interpretation ◽  
Integral Transform ◽  
The Laplace Transform ◽  
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In a recent paper [7] the author considered, among other things, the integral transform where is the fundamental solution of the heat equation There we gave a physical interpretation of the transform (1.1). Here we shall choose a slightly different interpretation, more convenient for our present purposes. If then u(O, t) = f(t). That is, the function f(t) defined by equation (1.1) is the temperature at the origin (x = 0) of an infinite bar along the x-axis t seconds after it was at a temperature defined by the equation .

scholarly journals Some recurrence relations and series for the generalised laplace transform

1960 ◽  
Vol 4 (3) ◽  
pp. 119-121 ◽  
Author(s):  
B. R. Bhonsle
Keyword(s):  
Laplace Transform ◽  
Recurrence Relations ◽  
The Laplace Transform ◽  
Recurrence Formulae ◽  
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The Laplace transformhas been generalised by Varma [4] by the relationwhich reduces to (1.1) when k = -m + ½ by virtue of the identityWe shall define πk, m, λ (p) by the relationThe object of this paper is to obtain some recurrence formulae and series for πk, m, λ (p) and to use them to obtain recurrence formulae and series for MacRobert's E-function.

Laser-Induced Heating of a Multilayered Medium Resting on a Half-Space: Part I—Stationary Source

1988 ◽  
Vol 55 (1) ◽  
pp. 93-97 ◽  
Author(s):  
R. Kant
Keyword(s):  
Laplace Transform ◽  
Half Space ◽  
Hankel Transform ◽  
Optical Recording ◽  
Time Intervals ◽  
Homogeneous Half Space ◽  
Multilayered Medium ◽  
Rule Application ◽  
The Laplace Transform ◽  
The Time Domain

Laser induced heating of a multilayered medium resting on a homogeneous half-space is considered. The transient heat transfer equation is solved by employing the Laplace transform in the time domain and the Hankel transform in the space domain (r direction). Numerical inversion of the Laplace transform is obtained by using a technique developed by Crump. For the time intervals of interest, inversion of the Hankel transform is obtained by the Simpson rule. Application to magneto-optical recording is discussed.

The function and associated polynomials

1951 ◽  
Vol 47 (2) ◽  
pp. 309-314 ◽  
Author(s):  
D. F. Lawden
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Difference Equations ◽  
Linear Differential Equations ◽  
Linear Difference Equations ◽  
Transform Method ◽  
Associated Polynomials ◽  
The Laplace Transform ◽  
Linear Differential ◽  
Image Position

A transform method for the solution of linear difference equations, analogous to the method of the Laplace transform in the field of linear differential equations, has been described by Stone (1). The transform u(z) of a sequence un is defined by the equation

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